Time dependent vector field

In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.

Definition

A time dependent vector field on a manifold M is a map from an open subset $\Omega \subset \Bbb{R} \times M$ on $TM$

$X: \Omega \subset \Bbb{R} \times M \longrightarrow TM$

$(t,x) \longmapsto X(t,x)=X_t(x) \in T_xM$

such that for every $(t,x) \in \Omega$, $X_t(x)$ is an element of $T_xM$.

For every $t \in \Bbb{R}$ such that the set

$\Omega_t=\{x \in M | (t,x) \in \Omega \} \subset M$

is nonempty, $X_t$ is a vector field in the usual sense defined on the open set $\Omega_t \subset M$.

Associated differential equation

Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:

$\frac{dx}{dt}=X(t,x)$

which is called nonautonomous by definition.

Integral curve

An integral curve of the equation above (also called an integral curve of X) is a map

$\alpha : I \subset \Bbb{R} \longrightarrow M$

such that $\forall t_0 \in I$, $(t_0,\alpha (t_0))$ is an element of the domain of definition of X and

$\frac{d \alpha}{dt} \left.{\!\!\frac{}{}}\right|_{t=t_0} =X(t_0,\alpha (t_0))$.

Relationship with vector fields in the usual sense

A vector field in the usual sense can be thought of as a time dependent vector field defined on $\Bbb{R} \times M$ even though its value on a point $(t,x)$ does not depend on the component $t \in \Bbb{R}$.

Conversely, given a time dependent vector field X defined on $\Omega \subset \Bbb{R} \times M$, we can associate to it a vector field in the usual sense $\tilde{X}$ on $\Omega$ such that the autonomous differential equation associated to $\tilde{X}$ is essentially equivalent to the nonautonomous differential equation associated to X. It suffices to impose:

$\tilde{X}(t,x)=(1,X(t,x))$

for each $(t,x) \in \Omega$, where we identify $T_{(t,x)}(\Bbb{R}\times M)$ with $\Bbb{R}\times T_x M$. We can also write it as:

$\tilde{X}=\frac{\partial{}}{\partial{t}}+X$.

To each integral curve of X, we can associate one integral curve of $\tilde{X}$, and vice versa.

Flow

The flow of a time dependent vector field X, is the unique differentiable map

$F:D(X) \subset \Bbb{R} \times \Omega \longrightarrow M$

such that for every $(t_0,x) \in \Omega$,

$t \longrightarrow F(t,t_0,x)$

is the integral curve $\alpha$ of X that satisfies $\alpha (t_0) = x$.

Properties

We define $F_{t,s}$ as $F_{t,s}(p)=F(t,s,p)$

1. If $(t_1,t_0,p) \in D(X)$ and $(t_2,t_1,F_{t_1,t_0}(p)) \in D(X)$ then $F_{t_2,t_1} \circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p)$
2. $\forall t,s$, $F_{t,s}$ is a diffeomorphism with inverse $F_{s,t}$.

Applications

Let X and Y be smooth time dependent vector fields and $F$ the flow of X. The following identity can be proved:

$\frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} (F^*_{t,t_0} Y_t)_p = \left( F^*_{t_1,t_0} \left( [X_{t_1},Y_{t_1}] + \frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} Y_t \right) \right)_p$

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that $\eta$ is a smooth time dependent tensor field:

$\frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} (F^*_{t,t_0} \eta_t)_p = \left( F^*_{t_1,t_0} \left( \mathcal{L}_{X_{t_1}}\eta_{t_1} + \frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} \eta_t \right) \right)_p$

This last identity is useful to prove the Darboux theorem.

References

• Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.